Traveling synchronized asymmetric two-waves in the propagation of the KdV and mKdV equations incorporating time-space dispersion terms

Joseph and Egri revised the standard Korteweg-de Vries equation by replacing its third-order space dispersion term by space-time dispersions aiming to adjust the wave speed and preserve frequency stability. The aim of the current study is twofold. First, it demonstrates that the Joseph-Egri equation exhibits dynamical behavior similar to the Boussinesq model, particularly in the propagation of synchronized asymmetric two-wave structures. Second, it presents and investigates a modified extension of the Joseph-Egri equation, inspired by similar modifications of the Korteweg-de Vries and Benjamin-Bona-Mahony equations. Effective schemes, including the trigonometric and hyperbolic rational functions method and the extended tanh-coth expansion method, are implemented to extract diverse explicit solutions for both models. These analytical results are further supported by 2D and 3D plots. The obtained findings reveal distinct physical structures for the Joseph-Egri and its modified counterparts. Specifically, the Joseph-Egri equation supports cusp soliton, bell-shaped soliton, and periodic concave-pattern solutions, while the modified Joseph-Egri equation admits kink soliton, periodic kink-pattern, and periodic concave-convex wave solutions. We believe the reported results will contribute to a deeper understanding of the synchronized propagation of bidirectional waves in various nonlinear media and related physical applications.